C172 pilot flying reported having multiple NMAC events with another aircraft while performing dual commercial flight training maneuvers.

Narrative

I was doing a Dual Commercial Training flight when about 45 minutes into the flight; me and my Flight Instructor noticed a DC-3 taking off from ZZZ but it was not a factor. At this time we were flying south and decided to turn north to see and avoid. After a few moments flying north; we noticed the DC-3 start flying north at the same altitude and gaining on us. We thought this was very strange and they got within a few miles of us. At this time both us and the DC-3 got a traffic advisory from ZZZ Tower. We decided to turn west and descend - they mimicked this turn and gained closer. We noticed this again and started a steeper turn to the east and descended. They mimic this again for a few seconds and then they ended up turning off. Since we saw they were trending away from us we started our maneuver again. About 20 minutes later we were doing a steep spiral and as we were turning around going east all of a sudden the DC-3 was right there. It happened so fast that I leveled out the wings and they were already not a factor.